lausen Function
المؤلف:
Arfken, G.
المصدر:
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.
الجزء والصفحة:
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9-8-2019
2565
lausen Function
Define
then the Clausen functions are defined by
![Cl_n(x)=<span style=]() {S_n(x)=sum_(k=1)^(infty)(sin(kx))/(k^n) n even; C_n(x)=sum_(k=1)^(infty)(cos(kx))/(k^n) n odd, " src="http://mathworld.wolfram.com/images/equations/ClausenFunction/NumberedEquation1.gif" style="height:98px; width:237px" /> |
(3)
|
sometimes also written as 
(Arfken 1985, p. 783).
Then the Clausen function 
can be given symbolically in terms of the polylogarithm as
![Cl_n(x)=<span style=]() {1/2i[Li_n(e^(-ix))-Li_n(e^(ix))] n even; 1/2[Li_n(e^(-ix))+Li_n(e^(ix))] n odd. " src="http://mathworld.wolfram.com/images/equations/ClausenFunction/NumberedEquation2.gif" style="height:56px; width:261px" /> |
(4)
|
For 
, the function takes on the special form
 |
(5)
|
and for 
, it becomes Clausen's integral
![Cl_2(x)=S_2(x)=-int_0^xln[2sin(1/2t)]dt.](http://mathworld.wolfram.com/images/equations/ClausenFunction/NumberedEquation4.gif) |
(6)
|
The symbolic sums of opposite parity are summable symbolically, and the first few are given by
for 
(Abramowitz and Stegun 1972).
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "Clausen's Integral and Related Summations" §27.8 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1005-1006, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.
Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 89-90, 2003.
Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, p. 27, 2004.
Borwein, J. M.; Broadhurst, D. J.; and Kamnitzer, J. "Central Binomial Sums, Multiple Clausen Values and Zeta Functions." Exp. Math. 10, 25-41, 2001.
Clausen, R. "Über die Zerlegung reeller gebrochener Funktionen." J. reine angew. Math. 8, 298-300, 1832.
Grosjean, C. C. "Formulae Concerning the Computation of the Clausen Integral 
." J. Comput. Appl. Math. 11, 331-342, 1984.
Jolley, L. B. W. Summation of Series. London: Chapman, 1925.
Lewin, L. Dilogarithms and Associated Functions. London: Macdonald, pp. 170-180, 1958.
Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981.
Wheelon, A. D. A Short Table of Summable Series. Report No. SM-14642. Santa Monica, CA: Douglas Aircraft Co., 1953.
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